1,357 research outputs found

    Scalable Semidefinite Relaxation for Maximum A Posterior Estimation

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    Maximum a posteriori (MAP) inference over discrete Markov random fields is a fundamental task spanning a wide spectrum of real-world applications, which is known to be NP-hard for general graphs. In this paper, we propose a novel semidefinite relaxation formulation (referred to as SDR) to estimate the MAP assignment. Algorithmically, we develop an accelerated variant of the alternating direction method of multipliers (referred to as SDPAD-LR) that can effectively exploit the special structure of the new relaxation. Encouragingly, the proposed procedure allows solving SDR for large-scale problems, e.g., problems on a grid graph comprising hundreds of thousands of variables with multiple states per node. Compared with prior SDP solvers, SDPAD-LR is capable of attaining comparable accuracy while exhibiting remarkably improved scalability, in contrast to the commonly held belief that semidefinite relaxation can only been applied on small-scale MRF problems. We have evaluated the performance of SDR on various benchmark datasets including OPENGM2 and PIC in terms of both the quality of the solutions and computation time. Experimental results demonstrate that for a broad class of problems, SDPAD-LR outperforms state-of-the-art algorithms in producing better MAP assignment in an efficient manner.Comment: accepted to International Conference on Machine Learning (ICML 2014

    Balancing lists: a proof pearl

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    Starting with an algorithm to turn lists into full trees which uses non-obvious invariants and partial functions, we progressively encode the invariants in the types of the data, removing most of the burden of a correctness proof. The invariants are encoded using non-uniform inductive types which parallel numerical representations in a style advertised by Okasaki, and a small amount of dependent types.Comment: To appear in proceedings of Interactive Theorem Proving (2014

    Compressive Network Analysis

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    Modern data acquisition routinely produces massive amounts of network data. Though many methods and models have been proposed to analyze such data, the research of network data is largely disconnected with the classical theory of statistical learning and signal processing. In this paper, we present a new framework for modeling network data, which connects two seemingly different areas: network data analysis and compressed sensing. From a nonparametric perspective, we model an observed network using a large dictionary. In particular, we consider the network clique detection problem and show connections between our formulation with a new algebraic tool, namely Randon basis pursuit in homogeneous spaces. Such a connection allows us to identify rigorous recovery conditions for clique detection problems. Though this paper is mainly conceptual, we also develop practical approximation algorithms for solving empirical problems and demonstrate their usefulness on real-world datasets

    FrameNet: Learning Local Canonical Frames of 3D Surfaces from a Single RGB Image

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    In this work, we introduce the novel problem of identifying dense canonical 3D coordinate frames from a single RGB image. We observe that each pixel in an image corresponds to a surface in the underlying 3D geometry, where a canonical frame can be identified as represented by three orthogonal axes, one along its normal direction and two in its tangent plane. We propose an algorithm to predict these axes from RGB. Our first insight is that canonical frames computed automatically with recently introduced direction field synthesis methods can provide training data for the task. Our second insight is that networks designed for surface normal prediction provide better results when trained jointly to predict canonical frames, and even better when trained to also predict 2D projections of canonical frames. We conjecture this is because projections of canonical tangent directions often align with local gradients in images, and because those directions are tightly linked to 3D canonical frames through projective geometry and orthogonality constraints. In our experiments, we find that our method predicts 3D canonical frames that can be used in applications ranging from surface normal estimation, feature matching, and augmented reality
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